A357698 a(n) is the sum of the aliquot divisors of n that are cubefree.

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 7, 1, 21, 1, 22, 11, 14, 1, 28, 6, 16, 13, 28, 1, 42, 1, 7, 15, 20, 13, 55, 1, 22, 17, 42, 1, 54, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 39, 17, 56, 23, 32, 1, 108, 1, 34, 41, 7, 19, 78, 1, 58, 27, 74, 1, 91, 1, 40

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n, d<n} A212793(d)*d.

a(n) = A073185(n) - (A212793(n)*n).

a(n) = 1 iff n is a prime.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2) - 1)/(2*zeta(3)) = 0.268262... .

Example

The divisors of 16 that are cubefree are {1, 2, 4}, and their sum is a(16) = 1 + 2 + 4 = 7.

Mathematica

f[p_, e_] := 1 + p + If[e == 1, 0, p^2]; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;; , 2]], # < 3 &], n, 0]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, 1 + f[i, 1] + if(f[i, 2] == 1, 0, f[i, 1]^2)); if(n==1 || vecmax(f[, 2]) < 3, s -= n); s};

Crossrefs

Cf. A004709, A073185, A212793, A293228.

Cf. A013661, A002117.

Sequence in context: A248029 A117552 A294886 * A069250 A294888 A364858

Adjacent sequences: A357695 A357696 A357697 * A357699 A357700 A357701

Keyword

nonn

Author

Amiram Eldar, Oct 10 2022

Status

approved